KOCH SNOWFLAKE, early and particularly elegant example of a fractal curve in mathematics. It was first proposed by Swedish mathematician Niels Fabian Helge von Koch in 1904.
A fractal refers to a geometric shape in which similar patterns recur at increasingly smaller scales. This quality is known as “self-similarity.” Fractal patterns are common in nature, such as in the structure of DNA, crystals, and galaxies. The Koch snowflake, like other recursive fractals, is unique as a geometrical shape with an infinite perimeter but a finite area, posing interesting problems for fractal geometry.
Fractal curves were introduced to mathematics by Karl Weierstrass in the 1870s as a way to challenge the assumption that continuous functions are always at least mostly smooth and therefore differentiable using ordinary calculus. A fractal curve is an example of a continuous curve that is essentially all corners and thus not differentiable at any point. Weierstrass’s examples of fractal curves were unwieldy and unlikely shapes that were difficult to visualize, especially without computers. Helge von Koch created his snowflake to serve as a clearer and more intelligible illustration of this principle.
To create a Koch snowflake, take an equilateral triangle and divide each side into thirds of equal length; create a new equilateral triangle using the middle portion of the subdivided side as its base; remove the base; repeat this process indefinitely with each outward-facing side of each new triangle. The triangles will branch out continuously into smaller and smaller triangles, creating the appearance of a snowflake.